Interactions,Actors,andTime: DynamicNetwork … · 2017-09-26 · nature, and actor-oriented nature of interpersonal interaction. We believe that this is a promising period to move

Citation: Stadtfeld, Christoph,and Per Block. 2017. "Interac-tions, Actors, and Time: DynamicNetwork Actor Models for Rela-tional Events." Sociological Sci-ence 4: 318-352.Received: March 10, 2017Accepted: April 9, 2017Published: May 15, 2017Editor(s): Jesper Sørensen, OlavSorensonDOI: 10.15195/v4.a14Copyright: c© 2017 The Au-thor(s). This open-access articlehas been published under a Cre-ative Commons Attribution Li-cense, which allows unrestricteduse, distribution and reproduc-tion, in any form, as long as theoriginal author and source havebeen credited.cb

Interactions, Actors, and Time: Dynamic NetworkActor Models for Relational EventsChristoph Stadtfeld, Per Block

ETH Zürich

Abstract: Ample theoretical work on social networks is explicitly or implicitly concerned with therole of interpersonal interaction. However, empirical studies to date mostly focus on the analysis ofstable relations. This article introduces Dynamic Network Actor Models (DyNAMs) for the study ofdirected, interpersonal interaction through time. The presented model addresses three importantaspects of interpersonal interaction. First, interactions unfold in a larger social context and dependon complex structures in social systems. Second, interactions emanate from individuals and arebased on personal preferences, restricted by the available interaction opportunities. Third, sequencesof interactions develop dynamically, and the timing of interactions relative to one another containsuseful information. We refer to these aspects as the network nature, the actor-oriented nature,and the dynamic nature of social interaction. A case study compares the DyNAM framework to therelational event model, a widely used statistical method for the study of social interaction data.

Keywords: social networks; DyNAM; relational event model; social interactions; network dynamics

INTERPERSONAL interactions, such as meetings and conversations, are at thecore of major social network theories. However, empirical research to date

mostly focuses on the study of durable interpersonal relations, such as perceivedfriendships and trust or advice relations, rather than on processes of interpersonalinteraction. In this article, we propose that the empirical study of interpersonalinteractions can offer insights that go beyond the study of stable relationships andcan complement our understanding of social network evolution and processes.To that end, we advance Dynamic Network Actor Models (DyNAMs) (Stadtfeld,Hollway, and Block 2017) to allow for the study of relational events of interpersonalinteraction.

Theoretically, the relationship between social interactions and durable interper-sonal relations is widely established. For example, Festinger, Schachter, and Back(1950) discussed how spatial proximity and thus opportunities for interaction affectthe formation of stable social ties, whereas Heider (1958:189–90) argued that thedistressing effect of interactions with friends in unbalanced situations may lead tothe dissolution of friendship. Diverse theories about social influence (e.g., Friedkin1998), about the structural advantage of network positions such as structural holes(Burt 2004) and structural embeddedness (Feld 1997), about homophily (Hustonand Levinger 1978), and about social exchange (Emerson 1976) are explicitly orimplicitly concerned with the role of interpersonal interaction.

From a theoretical perspective, interpersonal interactions can be understoodas reflections of individuals’ preferences, available interaction opportunities, andindividuals’ perception of the social network that they are embedded in. By ob-serving patterns of interactions through time, researchers can thus gain deeper

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Stadtfeld and Block Interactions, Actors, and Time

insights into the microlevel social mechanisms on the individual level that shapeand reflect complex social systems (Hedström and Bearman 2009). We refer to thisas the actor-oriented nature of interpersonal interaction.

Like stable social relationships, observations of interpersonal interactions are notindependent but reflect underlying social structures, such as reciprocity, clustering,or preferential attachment. Studying the dependence between instances of inter-actions may thus open insights into some of the most fundamental social networkmechanisms and, potentially, how those differ between individuals in differentnetwork positions. We refer to this feature as to the network nature of interpersonalinteraction.

When studying social interactions through time, another important dimensionof dependence unfolds that can reveal how sequences of interactions operate withinshort and longer time intervals. Reciprocal interactions, especially in communica-tion networks, may typically occur in short time sequences, reflecting turn-takingin conversations. Preferential attachment or clustering, on the other hand, mayoperate with a longer "process memory" in which events in the distant past may beconsidered more prominently by actors. Studying interpersonal interaction maythus open new insights into the relative timing of social network mechanisms. Werefer to this aspect as to the dynamic nature of interpersonal interaction. The DyNAMframework that we propose in this article is tailored to the dynamic nature, networknature, and actor-oriented nature of interpersonal interaction.

We believe that this is a promising period to move forward with the study of in-terpersonal interaction in social networks. The advent of electronic communication,social media, and human sensor technologies have brought about the possibility tocollect a wealth of fine-grained social interaction data. Interdisciplinary researchershave called for using such data to enrich the empirical toolbox of social scientists(Lazer et al. 2009; Kitts 2014). Besides these newly emerging, technology-drivenadvances, for many years social networks researchers have been collecting archivalnetwork data about, for example, political and financial relations that often containdetailed information about the timing of relational interaction. Thus, in a varietyof substantive disciplines, social network data are available as sequences of time-stamped traces of relational interaction to which we refer as relational event data, aterm coined by Butts (2008).

In general, data on interpersonal interactions consist of at least three elements1:the event time, the event sender, and the event receiver. One can think of examplessuch as meetings ("Person A meets person B at time T"), favors ("Person A doesperson B a favor at time T"), phone calls ("Person A calls person B at time T"), ore-mails ("Person A writes an e-mail to person B at time T"). In this article, we willrepeatedly return to the case of phone call sequences for illustrative purposes, andwe analyze one such data set in an empirical case study towards the end of thearticle.

This article develops DyNAMs (Stadtfeld et al. 2017) further, an actor-orientedclass of statistical models tailored to the analysis of time-stamped network data.While Stadtfeld et al. (2017) introduced DyNAMs for the study of undirected,durable coordination ties that are time-stamped, such as treaties between countriesor romantic relationships, in this article we advance DyNAMs for the study of

sociological science | www.sociologicalscience.com 319 May 2017 | Volume 4


directed event sequences. The inferential task of these DyNAMs is, broadly, toexplain who will send an event (event sender) to whom (event receiver) at whatpoint in time (event time). The explanatory parameters in the model can relate to,for example, events in the recent or more distant past (A calls B, after B called A),ongoing relations between individuals (A calls B, because A and B are friends), orfunctions of individual characteristics (A calls B, because A and B have attributesZ in common). The model attempts to closely reflect the network nature, actor-oriented nature, and dynamic nature of interpersonal interaction.

The DyNAM represents the network nature of interpersonal interaction byexplicitly modeling emerging network patterns over time as known from estab-lished network methods, such as the Exponential Random Graph Models (EGRMs)(Lusher, Koskinen, and Robins 2013) or Stochastic Actor-Oriented Models (SAOMs)(Snijders, van de Bunt, and Steglich 2010). Additional to the dependence on struc-tures in the network of previous interaction sequences, the model allows for takingthe larger, potentially complex social system into account. This can be representedby other, potentially coevolving networks of stable ties between individuals, com-mon associations to organisations (represented in two-mode networks), or nodal,dyadic, and global covariates.

The DyNAM is in the conceptual tradition of SAOMs, in which network changesare modeled as choices of individuals (Snijders et al. 2010) reflecting the model’sactor-oriented nature. Thus, it allows inference on individual preferences, theinteraction opportunities, and, potentially, individuals’ perception of the socialsystem. In the course of this article, the actor-oriented nature of the DyNAMis theoretically and empirically compared to the most widely used, tie-orientedmethod for the analysis of time-stamped network data, the Relational Event Model(REM) (Butts 2008).

The dynamic nature of the model is reflected by the model not only concerningthe order but also the absolute and relative timing of events. This is achieved byallowing the probability of events to vary by the amount of time that has passedsince previous events through specific time windows. As a consequence, thestatistical model needs to take periods of inactivity into account in a coherentway, as neglecting this can result in biased estimates. The importance of explicitlytreating time with reference to short-term and long-term dynamics is discussed andwill be illustrated in a detailed case study.

The remainder of this article is structured as follows. The DyNAMs for Re-lational Events section introduces Dynamic Network Actor models for relationalevents. In particular, it introduces the mathematical formulation that extends priorwork by Stadtfeld and Geyer-Schulz (2011) and Stadtfeld (2012) to multivariateprocess states and processes with right-censored intervals. It further revisits theactor-oriented modeling paradigm and contrasts the approach to tie-oriented mod-els such as the relational event model (Butts 2008). The Estimation section discussesestimation strategies for DyNAMs and REMs that are implemented in the R packagegoldfish. The Empirical Illustration of DyNAM and REM: Phone Calls in a StudentHouse section presents a straightforward empirical case study in which sequencesof phone calls between inhabitants of a student housing community are related toself-reported friendship ties. The DyNAM is further compared to the REM based



on likelihood-based model fit, interpretation of parameters, and computationalcomplexity. The Conclusions section discusses the future applicability and thepotential impact of the DyNAM.

Dynamic Network Actor Models (DyNAMs) for RelationalEvents

The model presented here is in the tradition of stochastic actor-oriented models(SAOMs; Snijders 1996) applied to relational event data (Stadtfeld 2012). Origi-nally, SAOMs have been developed to analyze the evolution of networks recordedas panel data. In recent years, however, SAOMs have been applied further assimulation models (e.g., Schaefer, Adams, and Haas 2013) and to cross-sectionaldata (Snijders and Steglich 2015). The model in this article extends and modifiesSAOMs to take into account the qualitatively different nature of relational eventsin comparison to network ties.2 In particular, we focus on relaxing assumptionsabout process myopia through time by allowing the inclusion of time-dependentstatistics while taking into account the additional complications that come with thisapproach. In the following sections, we refer to this adaptation of SAOMs to eventdata as DyNAMs, a terminology first used in Stadtfeld et al. (2017).

The Explanandum

A relational event is defined as a triplet ω that includes the event time tω , the eventsender iω, and the event receiver jω (in accordance with Butts 2008:159):

ω = (tω, iω, jω)

Observed relational event data consist of time-ordered sequences Ω = ω1, ω2, . . . of such triplets. The events in this series are typically not independent of oneanother, as they may contain the same individuals multiple times and relationalevents can, and often do, directly trigger one another (e.g., replying to or forwardinga received message).3 However, the dependence between events is generally not anuisance but is of direct interest to a network researcher, as this dependence canrepresent how people interact, react to others’ actions, and are affected by the sameexogenous factors.

The inferential task of the model is, then, to explain the timing, the sender,and the receiver of the relational event. Each of these can depend on previousevents, other relationships between actors, actor attributes, global variables (likethe time of the day) and much more. Jointly, all information that can influencewhich event transpires next (including information on past events) constitute theso-called "process state." We introduce this distinction between the event sequenceΩ (the "dependent variable") and the process state (the "independent variable") forconceptual reasons. As will become apparent in the course of this article, this cleardistinction simplifies modeling but without imposing any restrictions on the scopeof the model. At the same time, it makes a lot of modeling assumptions directlytractable.



Working with Multivariate Process States

The process state entails all information about the relevant world from the perspec-tive of the model at each time-point over the duration of the modeled period. Inprinciple, it is of arbitrary complexity. For the case at hand, we formally define theprocess state y(t) as a multivariate system at time t:

y(t) =(

x(1)(t), x(2)(t), . . . , z(1)(t), z(2)(t), . . . , A(1)(t), A(2)(t), . . .), (1)

y(t) ∈ Y

The elements x(·) represent networks or relational information of different kinds thatcan be expressed as matrices, the elements z(·) represent nodal or global covariatesthat can be expressed as vectors or scalars, and the elements A(·) represent nodesets that correspond to the dimensions of the matrices and vectors.

The process state elements include all relevant information about the currentprocess state and the relevant past (for example, all events sent from one individualto another up to time t, or all events sent within a specific time window before timet). It is worth emphasizing that by encoding all relevant information about the pastin the process state via different process state elements, the model probability ofobserving an event only depends on the process state and nothing else. For thisreason the model that we define can be understood as Markovian.

The process state can also include unmodeled, exogenously defined elements(e.g., advice relations between individuals or changing individual covariates).Dyadic variables and group affiliation will typically be expressed as networks,respectively. The elements x(·) can thus be one-mode as well as two-mode networks.All networks, dyadic attributes, and nodal attributes are related to one or twonode sets A(·). If group affiliation of individuals, for example, is represented by atwo-mode network x(·), two node sets A(·) will represent the two sets of individualsand groups. Nodal covariates z(·) could then be related to those node sets and, forexample, represent individuals’ gender and the age of a group.

As the process state is a function of time, all elements can be either constantor subject to change. The process state can change because of exogenous andendogenous processes. Endogenous processes are the ones that are described bya specific model, and exogenous processes are those that relate to informationupdates without being modeled explicitly. The most basic endogenous updatehappens after each realized event, when the network x(·)(t) that records all eventsup to time t is updated to include the event that just occurred. All parts of theprocess state that include information about the past event sequence are updatedafter an event transpired. An exogenous change of the process state might occurwhen, for example, the global variables are updated (e.g., seasonality or day time)or actors join or leave the network for unmodeled reasons. Process state elementsthat reflect certain aspects of the process history (e.g., indicate events that havehappened within the last 24 hours) will further update the process state with aspecified time-lag (e.g., 24 hours after an event has happened). Such time-laggedupdates will be discussed in the Right-Censored Intervals section.

Figure 1 illustrates an exemplary process state. It consists of five nodes connectedin two one-mode networks (x(1) and x(2)) and one two-mode network (x(3)) with



exogenous change

endogenous change

Process State

global variables

netw

orks

(on

e-m

ode

/ tw

o-m

ode)

nodal attributes

two-mode nodal attributes

x(1)

x(2)

x(3)

z(1) z(2)

z(3)

z(4)

z(5) z(6)

node sets

A(1) A(2)

Figure 1: An exemplary multivariate process state.

five additional nodes on the second mode. Thus, two node sets are part of theprocess state (A(1) and A(2)). One is indicated by circles, the other by squares.Further, the process state consists of two nodal attributes on each of the networkmodes and two global attributes (z(1), . . . , z(6)). Attributes are indicated by colors.

This conceptualization that includes all relevant model information in one pro-cess state simplifies working with increasingly complex network data, as practicedmore and more in recent years and exemplified by recent studies that simultane-ously analyze multiple networks (e.g., Boda and Néray 2015; Pál et al. 2016; Vörösand Snijders 2017) and, in particular, multimode or multilevel networks (Snijderset al. 2013; Wang et al. 2013; Lomi and Stadtfeld 2014; Hollway and Koskinen 2016).

An Actor-Oriented Model

Based on the process state y (for easier readability we omit the time indicator t whenpossible), we now define a probability model for the observed data Ω. In line withthe basic SAOM (Snijders 1996), and similar to previous approaches (Stadtfeld andGeyer-Schulz 2011; Vu et al. 2011; Perry and Wolfe 2013), the proposed model is atwo-step process with a linear predictor at its core. In the first step, the waiting timeuntil an actor becomes active to send an event is modeled. In the second step, giventhat a specific actor becomes active, the recipient of this event is modeled. Thesetwo steps are assumed to be conditionally independent given the process state.Applied to an empirical example, intuitively the first step models who will make



a phone call and at what time; the second step models who will be the receiver ofthat phone call.

Mathematically, the DyNAM models the waiting time between two subsequentevents. The waiting time is modeled as a composite Poisson process in which eachpossible event observation i→ j is associated with a Poisson rate ψij(y; θ, β). Thewaiting times between subsequent events are thus exponentially distributed. Underthe assumption that the two process steps (first, waiting time of a sender; second,discrete choice of a receiver) are conditionally independent given the process statey, we may express the first step as a Poisson process describing the activity of theactors and the second step as a probability distribution over all possible receivers:

ψij(y; θ, β) = τi(y; θ)︸︷︷︸select actor i

× p(i→ j; y, β)︸︷︷︸actor i sends event to j

. (2)

The waiting time τi of an actor i (step 1) depends on a parameter vector θ. Theprobability that an actor j is chosen as the event receiver (p(i→ j); step 2) dependson a parameter vector β. Thus, the waiting time for each event (i.e., what willhappen next) depends on the process state y and the two statistical parameters θ

and β guiding the two modeling steps. These two steps are now introduced indetail.

As suggested by Snijders (2005:232), the actor rates τi can be modeled with anexponential link function that can be specified with arbitrary real statistic functionsrk evaluating the process state.

τi(y; θ) = exp(θ0 + ∑

kθkrk(i, y)

)(3)

The parameter θ0 refers to the general tendency of each actor to send events throughtime (i.e., the intercept). The effect functions rk(i, y) relate to differences betweenactors regarding their attributes or positions in the process state.

The individual activity rates τi model the timing until an actor is "getting active"as the sender of an event (e.g., picks up the phone to call someone), given the processstate y. In Poisson models, the waiting time is exponentially distributed withparameters τi. In the language of event history modeling, τi(y; θ) can be understoodas the exponential hazard of actor i to get active—the hazard is independent of awaiting time δ.4 An exponential waiting time model5 is constructed by combiningtwo functions:

f (δ; y, θ, i) = τi(y; θ)e−τi(y;θ)δ,

S(δ; y, θ, i) = e−τi(y;θ)δ. (4)

The first function f it the exponential probability density of an actor i to be active attime δ. The second function S is one minus the exponential cumulative distributionfunction and thus refers to the probability of actor i not being active within a periodof length δ; in event history modeling, this function is called the survival function.The likelihood function (discussed in the Individual Activity Rates in the DyNAM



section) will later be defined as a combination of density functions f (of actuallyobserved events) and survival functions (of competing but nonrealized events).

In the models, multiple independent Poisson processes are competing in processstate y, evaluating which actor in the actor set A will send an event next. The overallwaiting time until any actor chooses to send an event is specified by an exponentialdistribution with parameter τ(y; θ).

τ(y; θ) = ∑k∈A

τi(y; θ)

The probability of a specific actor i to be the first to be active in the current state y isproportional to the actor rates (hazards).

p(actor i is active next; y, θ) =τi(y; θ)

τ(y; θ)

Two of the three elements of the event triplet—event time and event sender—arehereby determined in the first modeling step.

In the second modeling step, the third element, the event receiver, is determinedfollowing a multinomial probability distribution (McFadden 1974). One possibleinterpretation is that receivers j are determined as a choice of the event sender iamong all potential event receivers k given the process state y.

p(i→ j; y, β) =exp

(βTs(i, j, y)

)∑k∈A\i exp

(βTs(i, k, y)

) (5)

The statistics s(i, j, y) are functions of the process state that characterize the possibleevent i → j. For example, they might indicate that j has previously sent an eventto i (e.g., modeling the tendency to return a phone call) or that i and j are similaron some covariate (e.g., gender homophily). These statistics are in many wayscomparable to statistics (or effects) used in classic SAOM or ERGM studies (Lusheret al. 2013). These statistics will be further explored in the following section.

The basic model that we describe here is almost identical to the formulationof Snijders’ actor-oriented model (Snijders 2001)—a network change model wasinitially introduced for the analysis of actor-oriented processes in network paneldata. However, some adjustments have been made to reflect the qualitativelydifferent nature of interaction events and social ties, especially concerning multipleevents on the same dyad and the explicit treatment of time, which allows us torelax of the assumption of strict myopia. Further discussion will be given in theRight-Censored Intervals section.

The Model Statistics

Equations (3) and (5) include statistic functions r(i, y) and s(i, j, y) that measurepositions of nodes and ties in the process state. Thus, the selection of statistics in amodel determines what a relational event process depends on and, consequently,which hypotheses can be tested in empirical analyses. The goal is to express thatwithin certain structures the occurrence of an event will be more or less likely.



Micro-level mechanisms such as reciprocity, homophily, transitivity, and prefer-ential attachment can be tested (for an overview, see Kadushin 2012 and Rivera,Soderstrom, and Uzzi 2010) by operationalizing them as a statistic functions.

Stadtfeld et al. (2017) provide an overview of possible mathematical specifica-tions for the case of DyNAMs for undirected networks (coordination networks).The classification that they propose can be straightforwardly translated to directednetworks and model specifications proposed for actor-oriented models (Snijderset al. 2010; Block 2015; Ripley et al. 2016) and ERGMs (Snijders et al. 2006; Lusheret al. 2013), and REMs (Butts 2008; Marcum and Butts 2015) can be applied well.Given the vast literature on statistics for actor-oriented models, we only concep-tually introduce few model statistics that stand as examples for different effectclasses.

• Effects that depend on past events:

– The tendency to repeatedly send events to the same receiver within atime window. This effect can be operationalized as follows:

s1(i, j, y) = x(1)ij .

– The tendency to reciprocate an event that has been received within acertain time window:s(i, j, y) = x(1)ji .

Variable x(1) is a process state element of y that stores which events occurredwithin a certain time window as a weighted matrix. Matrix x(1) is updatedthrough events as well as through lagged events (referring to updates at theend of a time window).

• Effects that depend on other networks:

– The tendency to send events to a receiver that an actor is connected to inanother network:s3(i, j, y) = x(2)ij .

– The tendency to send events to a receiver that has the same affiliations ina two-mode network:s4(i, j, y) = ∑h∈A(2) x(3)ih x(3)jh .

Variable x(2) and x(3) are process state elements of y that store informationabout another one-mode network (e.g., friendship) and a two-mode network(e.g., joint membership in organizations). A(2) is the set of nodes in the secondnetwork mode. Both networks are examples of process state elements thatcould be exogenously updated or be modeled by another event process.

• Effects that depend on actor attributes:

– The tendency of actors to send more events if they score high on anindividual attribute (a rate effect):

r1(i, y) = z(1)i .



– The tendency to send events to receivers who score high on an individualattribute (a receiver choice effect):

s5(i, j, y) = z(1)j .

Variable z(1) is a process state element of y that stores information about anordered individual covariate (for example, individual attractiveness).

• Effects that depend on global attributes:

– The tendency of actors to send more events if a global variable is high:r2(i, y) = z(2).

The global variable z(2) could, for example, be one at daytime and zero atnight. A positive parameter would indicate that individuals are more activeduring the day.

• Effects that depend on a combination of the above:

– The tendency to choose individuals as receivers who have been chosenrecently by the sender and who the sender is indirectly connected to inanother network:s6(i, j, y) = x(1)ij ∑h∈A(1) x(2)ih x(2)hj .

This effect could model whether friends of friends who have been calledrecently are more likely to be called again. Flexible combinations of the typeof statistics above are possible.

In empirical analyses, these statistics can be calculated for each interval, com-binations of the sender and potential receivers, and each effect. An importantobservation is that these model statistics cannot just be used for the estimation ofDyNAMs but also for other types of network models such as the relational eventmodel (Butts 2008). The main difference between the DyNAM and the REM ishow the statistics are used to calculate probabilities, not the statistics themselves.6

Practical considerations about model estimation are discussed in the Estimationsection.

Right-Censored Intervals

In the introduction of the process state, it was considered that the process state maychange because of exogenous processes and time-lagged updates. An example forthe latter are cases in which nodes are allowed to show different activity patternsin the period after they have sent or received an event (i.e., explicitly modelingtime heterogeneity of the process.) Within a certain time window, the processstate would indicate that an actor is in a state of recent activity, allowing this toinfluence the subsequent probability to act. At the end of such an activity period,the process state is updated deterministically. Updates of the process state that donot follow endogenous events still constitute relevant model intervals—we referto such intervals as right-censored intervals. Further examples of right-censored



intervals are when exogenous information about unmodeled networks or nodalcharacteristics change, or when the set of actors is updated. In practical applicationswith many exogenous updates and detailed hypotheses about actor activity withintime windows, the number of right-censored events can clearly outnumber thenumber of intervals that end with an endogenous event.

Although not necessarily obvious, it is important to consider those intervals inthe estimation of models, especially for estimating time-dependent parameters. Forexample, assume we hypothesize that actors that were active within the past hourare more likely to send an event. In case we omit right-censored intervals in whichno event was sent within this hour, we would tend to over-estimate the importanceof this effect. This is because we would disregard information about all the cases inwhich actors did not act during this period—in a way, this would parallel an eventhistory model in which cases were deleted when the dependent variable is 0 buta specific independent variable is 1. Thus, considering right-censored intervals iscrucial for unbiased estimates of time-dependent parameters.

Right-censored and event intervals are illustrated in Figure 2 for a processof phone calls. The process state in this case consists of three nodes connectedthrough a network representing all call events in the past (upper row), a networkrepresenting all calls within the past day (second row), and a nodal attribute thatis subject to exogenous change (the color of nodes in the upper row). Nine timeintervals are defined by the process (I to IX) within which the process state isconstant. Change is brought about by endogenous updates that are due to phonecalls occurring (intervals I, IV, and V), by exogenous change of the nodal covariates(intervals II and VII), and by time-lagged updates of the second network one dayafter a phone call was observed (intervals III, VII, and VIII). The final interval IX isright-censored by the end of the observation period. Six out of nine intervals arethus right-censored. In the estimation of parameters, these right-censored intervalsneed to be considered as they entail relevant information about model parameters.

Revisiting the Actor-Oriented Paradigm

Although the actor-oriented paradigm for the analysis of network evolution is, bynow, firmly established, it is still instructive to review its foundational motivation,as discussed by Snijders (1996). Empirical tests of social theories often deduceassociations between variables and test these in an established statistical framework,but this is not the most direct test of these theories. As noted ibid.:

It would be preferable that the . . . deductions of the theory’s im-plications be integrated with the statistical model that is used for theempirical test. Such an integration leads to a statistical model that isitself a direct expression of the sociological theory. . . . The integrationof theoretical and statistical model is more complicated but it can lead tomore stringent theory development because it requires a completely ex-plicit theory and provides a much more direct test of the theory (Snijders1996:149).

If we assume that network ties or, in our case, relational events emanate fromactors that make conscious choices on whom to relate to, these decisions should be



call networknodal attribute

calls within past day

interval ended by eventinterval ended by exogenous change*

interval ended by time window update*

time

exogenousnode change

I II* III* IV V VI* VII* VIII*one day

exogenousnode change

IX*one day

one day

endogenousevent

endogenousevent

endogenousevent

time update

time update

time update

Figure 2: An exemplary process of call events that changes because of an endogenous process (call eventschange two network representations), a time window–related process (a network represents all phone callsthat were made within the past day), and an exogenous process (update of a nodal attribute). The processstate elements are shown in squares. Intervals that end because of exogenous changes or because of theupdate of time window representations (right-censored intervals) are marked with an asterisk.

explicitly modeled. This is the reason why the DyNAM, just like the SAOM, modelswhat makes actors more or less likely to act and, once they act, whom they interactwith. This explicit modeling of actions coming from individual actors originates inmethodological individualism.

While the motivation underlying an actor-oriented approach is straightforward,its consequences are not obvious and require some further discussion that are mosteasily illustrated by comparison to the tie-oriented REM. The differences are thatthe REM contains one modeling step (which event will happen next), whereas theDyNAM contains two steps (who will send an event, and to whom; see Figure 3).7

As a consequence, first, as opposed to the REM all modeling decisions are nestedwithin actors (Block et al. 2016). This means that the probability to observe events inthe REM only depends on its embedding in structures described by included effects.In the DyNAM, the probability to observe a specific event additionally dependson the other options an actor has to send events, as the different options for eventreceivers are compared by the sending actor. Thus, in an actor-based interpretationof the REM, a sender A decides whether it wants to send an event to a receiver B ornot, whereas in the DyNAM, actor A decides whether it wants to send an event toactor B, rather than to any other actors C, D, E, et cetera.

Second, the interpretation of model parameters differs between the REM and theDyNAM. In the DyNAM, the time process and change process can be interpretedindependently because those manifest in two separate modeling steps, whereas inthe REM, these processes are combined in one modeling step. This means that in theDyNAM, we can interpret the parameters in the time choice function as who will be



Actor-based model Tie-based model

1. Competing sender rates

2. Sender chooses receiver

1. Competing tie rates

Figure 3: Comparison of actor-oriented and tie-oriented models.

active next and when and parameters in the receiver choice function as, "given thata certain actor is active," who is most likely to be chosen as a receiver. In the REM,both statistics that relate to whom will be active next and what will that persondo must be interpreted net of the respective other model part. These differencesare exemplified in the empirical example, which also discusses additional modelevaluation criteria. Although these differences do not suggest that either model is"better," they show that the models differ inasmuch as the the nesting of decisionsin actors and the interpretation of parameters.

Estimation

This section presents the estimation of parameters from data for the actor-orientedDyNAM and the tie-based REM. Compared to the literature, the latter is extendedto incorporate right-censored intervals that stem from newly developed laggedwindow effects and exogenous process updates. All estimation routines have beenimplemented in the R package goldfish that is freely available for scientific use.8

The DyNAM consists of two subprocesses that are assumed to be conditionallyindependent, given the process state. The objective of the parameter estimationis to find the parameters θ and β under which the observed data are as likely aspossible. To this end, the likelihood of observed data needs to be defined. In orderto maximize this, the calculation of the first and second derivative (or a usefulapproximation of it) of these likelihoods is necessary as well.



Individual Activity Rates in the DyNAM

This section introduces the estimation procedure of the first sub process that wasdefined in Equation (3), in particular, of the maximizing parameter vector θ. Thisfirst modeling step is concerned with the actor rates that determine two of the threeelements of the event triplet: the time points tω and the event senders iω.

For each period observed one can now construct a likelihood function thatmakes use of the density function f (referring to actors who got active: observedevents) and the survival functions S (actors who could have been active but werenot: nonobserved events) in Equation (4). In right-censored intervals all potentialevents are non-observed. Let Ω be the set of all events, and Ω′ the set of right-censored intervals. The union set is denoted by Ω∗. For each element ωk in Ω∗,we define a time interval δωk = tωk − tωk−1 . This is the length of an interval that isended by an update of the process state. The set of actors that can act within a timeinterval is denoted by Aω and the piecewise constant process state by yω.

The partial likelihood Lτ to observe the timing and senders of a specific sequenceof events and right-censored intervals can then be expressed as:

Lτ(θ) = ∏ω∈Ω

( observed event︷︸︸︷τiω (yω; θ)e−δωτiω (yω ;θ)

nonobserved events︷︸︸︷∏

k∈Aω\iωe−δωτk(yω ;θ)

)︸︷︷︸

intervals ended by an event

∏ω′∈Ω′

∏k∈Aω′

e−δω′τk(yω′ ;θ)

︸︷︷︸right-censored intervals

= ∏ω∈Ω

(τiω (yω; θ) ∏

k∈Aω

e−δωτk(yω ;θ))

∏ω′∈Ω′

∏k∈Aω′

e−δω′τk(yω′ ;θ)

= ∏ω∈Ω

τiω (yω; θ)e−δω ∑k∈Aω τk(yω ;θ) ∏ω′∈Ω′

e−δω′ ∑k∈Aω′

τk(yω′ ;θ)

= ∏ω∈Ω

τiω (yω; θ) ∏ω∗∈Ω∗

e−δω∗ ∑k∈Aω∗τk(yω∗ ;θ)

This can be understood as the likelihood of a Poisson process with exponentiallydistributed waiting times and right-censored intervals. The formula expressesthat the likelihood of an observed time interval is for the left multiplier ("intervalsended by an event") the product of the exponential distribution density function(the hazard function) and the survival functions of all other rates (1 – distributionfunctions of the exponential distribution)—i.e., that actor i was getting active andnot any other actor k. The right multiplier ("right-censored intervals") expressesthat in the right-censored intervals no actor became active and it thus consists ofa product of individual survival rates. As is customary in maximum likelihoodestimation, its logarithm, but not the likelihood, is maximized:

log Lτ(θ) = ∑ω∈Ω

log(τiω (yω; θ)

)− ∑

ω∗∈Ω∪Ω′δω∗ ∑

k∈Aω∗

τk(yω∗ ; θ)

= ∑ω∈Ω

(θ0 + ∑

kθkrm(iω, yω)

)− ∑


k∈Aω∗

τk(yω∗ ; θ) (6)

The maximum of the likelihood function is reached when the first derivative ofthe likelihood with regard to each of the statistics rm equals zero. The first derivative



of the likelihood is defined as:

∂ log Lτ(θ)

∂θm= ∑

ω∈Ωrm(iω, yω)− ∑


k∈Aω∗

τk(yω∗)rm(k, yω∗) (7)

For the partial derivative of log(L) with respect to the intercept θ0, we defineconveniently rτ

0 (k, y) = 1 for any value of y and k. For this special case the firstderivative reduces to

∂ log Lτ(θ)

∂θ0= |Ω| − ∑

ω∗∈Ω∪Ω′∑

k∈Aω∗

δω∗τk(yω∗)︸︷︷︸expected number of events

(8)

where the first summand is the number of events observed and the second is theexpected number of events over all periods (it is sum over the expectation of asequence of Poisson distributions). The equation is zero and thus at maximumif the number of observed events equals the number of expected events over allevent periods and right-censored periods. The interpretation of Equation (7) for thenonintercept parameters is somewhat similar, but there the observed and expectedobservations are weighted with the statistics functions. Equation (8) calls to atten-tion that taking into account right-censored intervals is indeed important, as theright side is not invariant to the right-censored intervals in Ω′.The second derivative is defined in Equation (9).

∂ log Lτ(θ)

∂θm∂θn=− ∑


k∈Aω∗

τk(yω∗)rm(k, yω∗)rn(k, yω∗) (9)

The maximum likelihood estimate can then be found using, for example, an iter-ative Newton–Raphson procedure (Deuflhard 2004) based on the first and secondderivative shown above. Alternative estimation procedures could be applied, suchas a Fisher scoring method or other numerical procedures. The following sectionillustrates the Newton–Raphson logic in more detail.

Multinomial Receiver Choices in the DyNAM

The previous section explained partial likelihood inference related to the elementsiω (sender) and tω (time) of the event triplet ω = (iω, jω, tω). The second partiallikelihood is related to the determination of the receiver jω and finding a likelihoodmaximizing parameter vector β. Based on Equation (5), the likelihood of an eventsequence Ω can be formulated as

Lβ(β) = ∏ω∈Ω

exp(

βTs(i, j, y))

∑k∈A\i exp(

βTs(i, k, y))



with variables i = iω, j = jω, y = y(tω), A = Aω. The event log likelihood for oneevent ω is given by

log Lβω(β) = βTs(i, j, y)− log

(∑

k∈A\iexp

(βTs(i, k, y)

)). (10)

We propose to estimate the parameter vector with a Gauss/Fisher scoring methodas, for example, explained by Cramer (2003:36–38). A similar estimation routinethat explicitly calculated the Hessian rather than approximating it with the negativeinformation has been proposed by Stadtfeld (2012:45ff). The first derivative of alikelihood term with respect to parameter βm (the m-th element of the score vector)is

∂Lβω(β)

∂βm= p(i→ j; y, β)

(sm(i, j, y)− ∑

k∈A\ip(i→ k; y, β)sm(i, k, y)

)

= p(i→ j; y, β)

(sm(i, j, y)− sm(i)

). (11)

Function p(i→ j; y, β) was formulated in Equation (5). The first derivative of thelog likelihood for one event ω is defined as

∂ log Lω(β)

∂βm= sm(i, j, y)− sm(i). (12)

The abbreviation sm(i) expresses the expected realized statistic m of an actor i’sreceiver choice given parameter β. For example, this could be the expected numberof reciprocated relations that i is embedded in when choosing to send an event toone of the potential event receivers k in A with probability p(i→ k; y, β) of Equation(5).

The Fisher information matrix F is the inverse of the expected matrix of secondderivatives and can be utilized as an approximation of the latter in an iterativeNewton–Raphson optimization. The advantage is that the Fisher information matrixis typically easier to calculate. For a detailed introduction we refer to Cramer (2003).For the receiver choice submodel of the DyNAM, an entry with indexes m and n ofthe Fisher information matrix is defined as follows:

Fω;mn(β) = ∑k∈A\i

p(i→ k, y, β)

(sm(i, k, y)− sm(i)

)(sn(i, k, y)− sn(i)

). (13)

An updating step in the Fisher scoring method of a parameter βt in iteration tuses the scores as well as the Fisher matrix to iteratively approximate the maximum



likelihood estimate β.

βt+1 = βt + F(βt)−1 ∂ log L(βt)

∂βt , (14)

F(βt) = ∑ω∈Ω

Fω(βt)

log L(βt) = ∑ω∈Ω

Lω(βt)

Relational Event Model

The relational event model introduced by Butts (2008) is essentially a Poissonmodel that is equivalent to the one presented in the Individual Activity Rates in theDyNAM section. It is, however, somewhat more complex than the estimation ofactor activity rates because the competing Poisson rates are defined for all sender–receiver combinations. In a network with n active nodes, n Poisson rates will becompeting in the DyNAM but n(n − 1) Poisson rates in the REM. The rates foreach sender–receiver combination i, j can be defined based on a linear function(see Butts 2008:166) with a fixed intercept parameter γ0 and additional parametersweighting structures in the process state.

λij(y) = exp(γ0 + ∑

kγksλ

k (i, j, y))

(15)

The log likelihood then looks similar to Equation (6). Functions sk are modelstatistics like the ones introduced in The Model Statistics section. Their effect onthe process dynamics is described by parameter vector γ that is subject to statisticalinference. Other than in its original formulation in Butts (2008), we incorporateright-censored intervals throughout the process and thus allow exogenous as wellas lagged process changes.

log Lλ(γ) = ∑ω∈Ω

(γ0 + ∑

kγksλ

k (yω, iω, jω))− ∑


k,l∈Aω∗

λkl(yω∗) (16)

The implemented estimation routine is a Newton–Raphson optimization andresembles the one presented in the Individual Activity Rates in the DyNAM section.Only the computational and data complexity is higher, as more Poisson rates arecompeting. The log likelihood log Lλ(γ), however, is the complete process loglikelihood whereas in the case of the actor-oriented models, the complete processlog likelihood is determined as a sum of the two partial log likelihoods log Lτ(θ)

and log Lβ(β) (Eqs. [6] and [10]).

Partial Likelihood Inference and Semiparametric Models

One key advantage of parametric models such as the ones discussed above (Dy-NAM and REM) is that it is possible to use them for process simulations. Thereby,one can, for example, generate simulation-based goodness-of-fit criteria as knowfrom ERGMs (Lusher et al. 2013) and SAOMs (Lospinoso 2012).



An alternative to parametric models such as the ones shown above are semipara-metric models (achieved by integrating out the rates and focusing on the estimationof ordered models and a subset of non–time-dependent effects) (Butts 2008; Stadt-feld et al. 2017). This is reasonable for processes in which no right-censored intervalsexist but is more problematic if this cannot be assumed.

Another alternative to full parametric models are Cox models (Cox and Oakes1984). A theoretically plausible feature of Cox models is that distribution assump-tions about the hazards, such as exponentially distributed waiting times in thecase of DyNAM and REM, are not necessary. Rather, under the assumption ofproportional hazards, Cox models are unconcerned about the actual underlyingdistribution. Perry and Wolfe (2013) proposed proportional hazard estimation forevent data. This is reasonable as long as the model still assumes piecewise stableprocess states as in this article; this means that Cox models also need to considerright-censored intervals. Once effects are taken into account that, for example,exponentially weight the past,9 the proportional hazards assumption is violatedand it is unclear in which biases this would result.

Empirical Illustration of DyNAM and REM: Phone Calls ina Student House

The Social Evolution Data

As an empirical illustration, we analyze the "Social Evolution" data set of theMassachusetts Institute of Technology Media Lab, a rich data set of 84 individualsliving in a U.S. student house. The data were originally collected to assess the"health state" of a social community, particularly by observing changes in physicalhealth but also health-related norms (Madan et al. 2012). Here, we use the newlydeveloped goldfish software to analyze a sequence of phone calls between theinhabitants (automated, time-stamped data collection) and relate those to change ina friendship network (based on a longitudinal survey at two points in time). Thisreflects our introductory comment about how interaction studies may complementnetwork studies about stable relations, such as friendship.

Figure 4 shows the phone call network within the first two months of the datacollection (black arrows, excluding isolates). Self-declared friendship relations areillustrated by blurry blue ties in the background. Node color relates to the gradetype of individuals who are students at different stages of their respective studyprograms. On visual inspection it seems that friendship relations are related tophone call interactions and that the network exhibits a certain level of grade-typehomophily. We make use of the DyNAM to test the effects of such descriptivefindings in a dynamic framework. Not only do we present results of the DyNAMbut we compare those to the relational event model (Butts 2008), in which right-censored intervals are additionally taken into account.

The process state that we define to model these data consists of six elements:

• a weighted network that measures who has called whom in the past ("callNet-work")



Figure 4: The call network (without isolates) of the first two months of data collection. Friendship relationsare indicated by shadowed ties in the background. The node color relates to the grade type of individuals;missing values are indicated by nodes with a central dot.

• a weighted network that measures who has called whom in the past hour("callNetworkPastHour")

• a network that expresses who has nominated whom as a friend ("friendship")

• an actor variable that expresses the floor on which individuals live in thestudent house ("floor")

• an actor variable that expresses the grade type of individuals ("gradeType")

• an actor set representing 84 individuals

The model statistics refer to these process state elements. The first three elementschange over time whereas the last three are constant.10 The second element ("call-NetworkPastHour") is changed through events (ties are increased by one) as wellas through lagged updates (values are decreased one hour after a phone call tookplace). The short names in parentheses above will be used in the following modelspecification as well as in the Results section.



Empirical Model Specifications

The specification of the empirical case study is kept straightforward and consistsof effects that relate to the general activity of actors (the DyNAM rates) and thepropensity to choose certain receivers (the DyNAM choice model). In the REM,all parameters simultaneously affect the timing and sequence of events, without aseparation into two submodels. Mathematical details about the model specificationsare provided in Appendix A of the online supplement. The naming conventionsare in line with the short names of effects in the SIENA software (Ripley et al. 2016)for the estimation of actor-oriented models for network panel data.

Five effects are concerned with the general activity of actors in both models. Incase of the DyNAM, these five effects specify the sender rate shown in Equation (3).The most basic one is an intercept that models

1. the general activity of actors to make phone calls ("intercept")

We further test how the position of nodes in the process states relates to theiractivity, in particular whether the propensity to make a phone call it is affected bythe following:

2. having made phone calls within the past hour ("egoX recentCallsSent")

3. having received phone calls within the past hour ("egoX recentCallsReceived")

4. having more friends ("outdegreeX friendship")

5. having called more people in the past ("outdegreeX callNetwork")

Eleven effects are concerned with the choice of a call receiver. These effectsconstitute the specification of the receiver choice submodel in Equation (5). Theytest the following:

6. whether individuals tend to call those they have called before ("outdegreecallNetwork")

7. whether individuals tend to call those they have called in the past hour("outdegree callNetworkPastHour")

8. whether individuals tend to call those who have called them before ("reci-procity callNetwork")

9. whether individuals tend to call those who have called them in the past hour("reciprocity callNetworkPastHour")

10. whether individuals tend to call those who have been called by many others("inPop callNetwork")

11. whether individuals tend to call those who have been nominated as a friendby many others ("inPop friendship")

12. whether individuals tend to call those who have been called by past receiversof their own past phone calls ("transitivity callNetwork")



13. whether individuals tend to call friends of their friends ("transitivity friend-ship")

14. whether individuals tend to call others who live on the same floor ("sameXfloor")

15. whether individuals tend to call others who are in the same grade type("sameX gradeType")

16. whether individuals tend to call their friends ("X friendship").

Effects 2, 3, 7, and 9 are related to changes within a one-hour time window.Effects of that kind (e.g., the tendency of actors to make phone calls if they havecalled someone recently) were introduced as time window effects in Stadtfeld et al.(2017).

Results

Results of the case study are presented in Table 1. The leftmost column includesestimates of the DyNAM rate model, the center column those of the DyNAMreceiver choice model, and the right column the estimates of a REM that wasspecified in the same way. Below the table, the log likelihoods are shown as well asthe CPU time that was needed for the estimation. In case of the DyNAM, the jointlog likelihood of both sub models is printed below the center column. The "interval"log likelihood relates to the full models as described in this article, whereas the"sequence" log likelihood relates to a simpler model in which only the order of1,092 phone call events was modeled without taking into account timing and right-censored intervals. The Biases When Ignoring Right-Censored Intervals discussesbiases in such sequence models in more detail.

In the DyNAM, both submodels can be interpreted independently because itassumes conditional independence of the two submodels given the process state.

The five DyNAM rate effects (effects 1–5) can be interpreted as "what affectsthe tendency of individuals to make phone calls," irrespective of who they arecalling. The intercept effect models the general tendency of individuals to makephone calls. The intercept can be loosely interpreted as follows: Individuals whohave not made and received calls recently, have no friends, and have in generalnever made a phone call (the model statistics of parameters 1–4 are thus zero) havean expected waiting time until they make a phone call of 10.7 days.11 The otherfour rate parameters are positive and can thus be interpreted as all increasing thetendency of individuals to make phone calls (and thus decreasing the expectedwaiting time between subsequent calls). For example, our model results suggestthat individuals who have made and received five phone calls within the past hour,have five friends, and have called five people ever before will only have an expectedwaiting time until the next phone call of 13 minutes.12 DyNAM rate effects sizescan be interpreted in terms of relative probabilities. For example, one could looselystate that with each additional person someone has called within the past hour,the expected waiting time decreases by about 45 percent, all else being equal.13 Atthe end of the one-hour window following a phone call a right-censored intervalconcludes the higher activity period of an actor.



Table 1: Results of the case study using DyNAM and REM.

DyNAM REM(1) (2) (3)

# Effects θ (s.e.) β (s.e.) γ (s.e.)

1 intercept −13.74† −14.78†0.05 0.14

2 egoX recentCallsSent 0.59† 0.55†0.04 0.03

3 egoX recentCallsReceived 0.53† −0.27†0.05 0.05

4 outdegreeX friendship 0.03† −0.04†0.00 0.01

5 outdegreeX callNetwork 0.26† −0.030.01 0.02

6 outdegree callNetwork −4.09† −4.97†0.16 0.13

7 outdegree callNetworkPastHour −2.06† 0.79†0.19 0.11

8 reciprocity callNetwork 0.24 0.38†0.14 0.11

9 reciprocity callNetworkPastHour 4.01† 4.67†0.32 0.13

10 inPop callNetwork −0.10∗ 0.010.05 0.03

11 inPop friendship −0.17† −0.06†0.02 0.01

12 transitivity callNetwork 0.30∗ −0.16∗0.12 0.07

13 transitivity friendship 0.02 0.14†0.04 0.02

14 sameX floor −0.24∗ −0.98†0.12 0.08

15 sameX gradeType 0.13 −0.17∗0.12 0.08

16 X friendship 2.07† 1.52†0.18 0.12

Log likelihood (submodel) −14, 165.94 −1, 319.38 −14, 800.81Log likelihood (interval) −15, 485.32 −14, 800.81Log likelihood (sequence) −4, 987.42 −4, 513.31CPU time (seconds) 18.11 2, 111.69

Column (1) includes the DyNAM sender rate, column (2) the DyNAM receiver choice, and column (3) theREM tie rate. Standard errors are reported below the estimates. The data set includes 1,092 phone call eventsand 1,093 right-censored intervals between 84 actors. The likelihood ratio per interval is 1.368 (REM overDyNAM, considering both events and right-censored intervals).∗ p < 0.05, † p < 0.01



The eleven receiver choice effects (effects 6–16) can be interpreted as "whataffects the choice of who will be called". Given that a certain individual makesa phone call at a certain point in time (modeled by the rate), the results showthat it is very likely that a receiver will be chosen that has been called before (thenegative "outdegree callNetwork" effect 6 suggests that individuals keep theiroutdegree low), particularly if the receiver has already been called within the pasthour (negative "outdegree callNetworkPastHour" effect 7). Further, we find noevidence that individuals tend to call those who have called them before at anypoint in the past ("reciprocity callNetwork" effect 8); however, there is a strong effectif they have been called by that person within the past hour (positive "reciprocitycallNetworkPastHour" effect 9). There is strong evidence that people tend to calltheir friends ("X friendship" effect 16). The "transitivity callNetwork" effect ispositive, which means that individuals tend to call those who they are connectedto through one or several two-paths in the phone call network. There is, however,no evidence that being connected through a friendship two-path increases theprobability of choosing a receiver ("transitivity friendship") and thus calling friendsof friends who are not friends (we control for the direct friendship effect already).The two effects relating to indegree popularity in the call network and in thefriendship network (10:"inPop callNetwork"; 11:"inPop friendship") are negative,which indicates that the process is characterized by tendencies towards a balancedindegree distribution rather than a strong centralization. Only one of the two nodaleffects is significant, and we find evidence for calling people who live on a differentfloor (14: "sameX floor"). An interpretation as relative probabilities is again possible.For example, our findings suggest that individuals who are friends are 7.9 timesmore likely to call each other, all else being equal.14 It is important to note that theinterpretation of effects within the receiver choice model are all net of each other.For example, we descriptively see that on an aggregated level, people have a highertendency to call others who live on the same floor. Once we control, however, forthe other parameters such as those that model that friendship networks are strongerwithin the floor network, we find a negative residual effect for floor homophily.

The interpretation of REM parameters is somewhat more involved as all param-eters affect the waiting times between subsequent events as well as the tendencythat an event of a specific sender–receiver pair is occuring next. The REM is tie-oriented, which means that all tie Poisson rates are competing at all times. Inan actor-oriented model, the receiver choice effects are nested within actors andalternatives are thus only compared for the actor that actually decides to make aphone call. Some estimates of the REM indeed have a different sign as comparedto the DyNAM, which illustrates the differences in interpretation. For example,the "outdegreeX friendship" effect is negative. This could relate to the significant"transitivity friendship" effect in the model: individuals with many friends tendto be embedded in more transitive friendship clusters within which phone callsare more likely and more frequent. The individual "outdegreeX friendship" effectthus potentially balances out the structural "transitivity friendship" effect. In theDyNAM, both effects can be interpreted independently, as they are part of differentsubmodels. We could thus straightforwardly conclude that individuals with manyfriends are more likely to make phone calls—an observation that is supported by



descriptive analyses of the data set but is difficult to infer from the REM results.Similar to the DyNAM, we can also interpret parameters in terms of relative prob-abilities; however, now we need to take the tie-oriented nature into account. Theprobability that the next event in the sequence will be a phone call between any pairof friends is 4.6 times more likely than a phone call between any pair of individualswho are not friends.15

Overall, it can be said that the REM has a significantly better fit regarding loglikelihood (higher values are better). The likelihood ratio per event is 1.37, whichmeans that the REM is on average 37 percent more precise for any event. Thisdifference is explained by the fact that in the REM, all effects model waiting timesand the event sequence. Further, all tie rates are defined as competing waitingtime processes and their evaluation is not nested within actors. The two submodelsof the DyNAM are either concerned with actor waiting times or with with thechoices of event receivers. A drawback of the higher precision of the REM is thehigher computational complexity, which led to a CPU time that is a factor 60 higher.The following two sections explore the model likelihoods and the computationalcomplexity in more detail.

Comparison of Event Likelihoods

By zooming in on the likelihood of single events, researchers can gain deeperinsights into where the DyNAM performs well and where there are differencesbetween the actor-oriented DyNAM and the tie-oriented REM. Figure 5 showsthe likelihood ratios for each of 1,092 events in two submodels that ignore right-censored intervals and are just concerned with ordered event sequences (like theearlier models proposed by Butts 2008 and Stadtfeld 2012). As some of the likelihoodratios (in particular for the case of unlikely events) have extremely high valuesof more than a thousand, the likelihood ratios were logged below and above thebaseline. It can be seen that the REM performs better in most events. The averagelikelihood ratio of non-censored intervals (intervals ending with a phone call event)is 1.95 per event in favor of the REM, whereas the likelihood ratio of right-censoredintervals is 1.09 in favor of the DyNAM. A closer look at two outliers better explainsunder which circumstances each model performs better.

Outlier 1 in Figure 5 is an event that is considered rather unlikely by bothmodels (only 8 times more likely than choosing a random tie in the REM andquite impossible in the DyNAM, with just 0.07 of the random choice likelihood).The event likelihood of the REM, however, is more than 1,000 times higher thanthe DyNAM event likelihood. The event describes a phone call at 3:30 a.m. ona Monday morning between two individuals who have been calling each otherregularly (but not within the last hour). The DyNAM probability of the call senderto make a phone call is extremely small because five other individuals have beencalling each other within the past hour. The most likely event from a DyNAM’spoint of view would be that one of those five makes another phone call, given that"egoX recentCallsSent" and "egoX recentCallsReceived" are the dominant effectsin the sparsely specified DyNAM rates. The REM, however, takes all effects intoaccount when modeling waiting times between events. It can therefore factor into



1 100 300 500 700 900 1092

53

03

57

DyNAM

REM

outlier 1

outlier 2

Figure 5: Likelihood ratios (logged for better readability) per event in an ordered model that ignores right-censored intervals and just models sequences. Events above the 0 line favor the REM, events below favorthe DyNAM. Two outliers are exemplarily discussed in the text. The dashed line shows for each event theaverage likelihood ratios of all prior events.

the waiting time model that the two individuals are friends and have called eachother before. Those two parameters are only considered in the DyNAM conditionalon the probability that the actual sender makes a phone call, which is extremelyunlikely given recent activity.

Outlier 2 in Figure 5 is an event that is considered rather likely in both models(149 times more likely than choosing a random tie in the DyNAM). The DyNAMevent likelihood is 42 times higher than the REM event likelihood. Here, theDyNAM rates put a lot of weight on the actual call sender, as this person has a veryhigh outdegree in the friendship network and in the call network. Both effects areassociated with a high activity of actors. In case of the REM, these effects negativelycontribute to the activity of actors net of all other effects (e.g., the "X friendship"effect).

The comparison of event likelihoods can help researchers improve the fit oftheir empirical models. Our exemplary analyses, for example, showed that thespecific, exemplary DyNAM performs poorly in case the three rate parameters do



DyNAM

REM

Rateeffects

Effect 1 Effect 2 Effect 3

Figure 6: Information used (change of statistics given an event between any pair of actors) for the estimationof DyNAM and REM, respectively. The matrix cells of model statistics of the observed event are markedwith a frame.

not explain activity but rather do preexisting network structures. One could nowconsider extending the model so that individual rates gain a better precision in caseof 3:00 a.m. phone calls between old and regularly interacting friends.

Comparison of Computational Complexity

Table 1 shows that the relational event model needs significantly more CPU time(by a factor of about 60). Some differences may stem from a more elaborate imple-mentation of the DyNAM using a Fisher scoring method; however, the major partof the differences is related to the fact that within the REM framework, a higheramount of information is processed. Figure 6 visualizes this difference: in theDyNAM, a number of parameters that do not vary within actors (e.g., outdegree inthe friendship network) are used to specify the rates. Of those parameters that varywithin actors (e.g., transitive closure of different choice alternatives), only thoseof the chosen sender are taken into account. In contrast, within the REM rates, allmodel statistics of all actors are compared for each event. Roughly speaking, thiscorresponds to a computational complexity that is N times larger, where N is thenumber of actors (84 in the example). Further complications arise because of highermemory demands. However, those do not affect the reported CPU times in Table 1because the data set chosen could be processed with the RAM of a single computer.In general, one can conclude that because of a lower computational complexity,DyNAMs are better suited for large data sets with a high number of actors. Theestimation routines for both models are parallelized within the goldfish packageso that this potential drawback of the tie-oriented REM can be overcome and modelselection can often be made based on theoretical assumptions.



Biases When Ignoring Right-Censored Intervals

We argue that it is important from a statistical perspective to consider right-censoredintervals when modeling relational event sequences and that disregarding themshould lead to biased estimates. But what are the biases induced in our examplemodels? Table 2 compares estimates of the full DyNAM rate model (left) andestimates that just take into account the order of events given the process stateat actual event times. In both models, the parameters are clearly significant andpoint into the same direction. However, the absolute values of the two actor effectsthat relate to the time window effects are larger in the model that ignores right-censoring, whereas the other two effects are very similar. Comparing estimates ofexponential family models across models and across data has to be taken with agrain of salt, and we thus refrain from substantively interpreting these differences.It is notable, however, that the absolute change of the parameters is in line with ourearlier expectations: when we ignore modeling lagged updates that describe timewindows in which actors are potentially more active (one hour in case of our casestudy), we will overestimate time window–related effects.

The biases are a lot more pronounced when comparing an ordered relationalevent model that ignores right-censored intervals with a full relational event modelas proposed in this article. This comparison is shown in Table 3. Of 15 effects,nine would have been interpreted differently in a model ignoring right-censoredintervals, either because the parameter sign or level of significance changed. Theinterpretation of the differences is not straightforward because the interpretation ofeach effect in this model is net of all others. The biased effects that relate to structuralchange are now much closer to the DyNAM receiver choice sub model (see Table 1),which is not concerned with timing but solely models receiver choices once an actorconsiders making a phone call. The effects that are modeled as DyNAM rates arevery different, however, and two of those have a different sign ("outdegreeX"). Itis thus unclear how to interpret the estimates of the REM that is ignoring right-censoring, but we can conclude that the induced biases are substantial.

Conclusions

Interpersonal interactions are at the core of major social network theories. However,empirical research to date mostly focuses on the study of stable relations (such asfriendship, kinship, and collaboration) rather than on processes of interpersonalinteraction (such as contact opportunity, communication, and knowledge exchange).We argued that the availability of new data collection techniques and advances instatistical network models may lead to a paradigm shift in how empirical researchaddresses questions about the dynamics of social networks by focusing on the dy-namic patterns of interpersonal interaction. In an attempt to contribute to this newperspective, this article introduced DyNAMs for the study of directed interactionsequences through time.

DyNAMs are relational event models in the spirit of the seminal work by Butts(2008) but build upon the actor-oriented paradigm of Snijders (1996), a combinationof two research lines that was discussed by Stadtfeld and Geyer-Schulz (2011).



Table 2: Comparison of the bias of the DyNAM rate induced by ignoring right-censored intervals.

(1) (2)Effects θ (s.e.) θ′ (s.e.) ∆

intercept −13.74†0.05

egoX recentCallsSent 0.59† 1.14† 0.550.04 0.05

egoX recentCallsReceived 0.53† 1.29† 0.750.05 0.05

outdegreeX friendship 0.03† 0.02† 0.010.00 0.00

outdegreeX callNetwork 0.26† 0.23† 0.030.01 0.01

Column (1) includes a continuous-time DyNAM rate model, column (2) a sequential model ignoringright-censoring. Standard errors are below estimates.∗ p < 0.05, † p < 0.01

DyNAMs for the study of (undirected) coordination network have been introducedin Stadtfeld et al. (2017). We argued that our model fulfills three core criteria thatwe believe are useful both theoretically and methodologically. First, the actor-oriented nature of the model aligns with a variety of theoretical explanations thatconsider individual preferences, available interaction opportunities, and, potentially,individuals’ perceptions about their position in the social system. Second, theDyNAM is a network model that can take into account how social interactionsare embedded in a social system with a complex structure that consists of, forexample, various dynamically changing networks and nodal and global covariates.Third, we elaborated on how to study the absolute and relative timing of eventsequences and explained under what circumstances models that take into accounttime windows and exogenous changes return unbiased estimates. This articlepresented the theoretical foundations of DyNAMs, their mathematical formulation,a numerical estimation method, and a detailed case study in which we compared itto the tie-oriented REM (Butts 2008).

We believe that this is a good time for proposing a new method to enrich thetoolbox of social scientists for the analysis of network data, given the constantlygrowing interest in analyzing relational events over the past decade. Recent ex-amples include the study of communicational dynamics of animals (Tranmer et al.2015) and emergency responders (Butts 2008), brokerage and receiver choice in com-munication networks (Quintane and Carnabuci 2016; Stadtfeld, Geyer-Schulz, andAllmendinger 2011), e-mail communication in organizations (Perry and Wolfe 2013),interaction within teams (Leenders, Contractor, and DeChurch 2016), exchangeof patients between hospitals (Kitts et al. 2017; Vu et al. 2017), and collaborationon online platforms (Vu et al. 2011). Further, international conflict relations wereanalyzed in a relational event framework (Lerner et al. 2013).



Table 3: Comparison of the bias of the REM induced by ignoring right-censored intervals.

(1) (2)Effects γ (s.e.) γ′ (s.e.) ∆

intercept −14.78†0.14

egoX recentCallsSent 0.55† 0.17† 0.380.03 0.05

egoX recentCallsReceived −0.27† 0.19∗ 0.460.05 0.08

outdegreeX friendship −0.04† −0.04† 0.010.01 0.01

outdegreeX callNetwork −0.03 −0.11† 0.080.02 0.02

outdegree callNetwork −4.97† −5.44† 0.470.13 0.14

outdegree callNetworkPastHour 0.79† −1.95† 2.740.11 0.14

reciprocity callNetwork 0.38† 0.09 0.280.11 0.10

reciprocity callNetworkPastHour 4.67† 2.53† 2.140.13 0.17

inPop callNetwork 0.01 −0.13† 0.130.03 0.03

inPop friendship −0.06† −0.10† 0.050.01 0.01

transitivity callNetwork −0.16∗ 0.22† 0.380.07 0.08

transitivity friendship 0.14† 0.16† 0.020.02 0.02

sameX floor −0.98† −0.43† 0.550.08 0.08

sameX gradeType −0.17∗ −0.19∗ 0.030.08 0.09

X friendship 1.52† 1.41† 0.110.12 0.13

Column (1) includes a continuous-time REM, column (2) a sequential model ignoring right-censoring.Standard errors are below estimates.∗ p < 0.05, † p < 0.01

A significant part of this article was concerned with contrasting the DyNAM tothe tie-oriented REM. We explained the theoretically different assumptions of actor-and tie-oriented network models, building upon the cross-sectional comparisonof Block et al. (2016). The DyNAM consists of two sub processes. One modelstiming of events and actor activity and one the choice of event receivers. In REMs,



all parameters in the model contribute to the modeling of timing, sender, and re-ceiver, and thus it has clear advantages over the DyNAM in a likelihood-basedmodel comparison. The fact that REM parameters simultaneously model timingand position of events, however, means that the parameters are typically moredifficult to interpret. We showed that ignoring timing of events can potentially leadto significant biases of parameter estimates. In the empirical case study, we couldillustrate that interpretation of DyNAMs is rather straightforward, which facilitatesthe testing of hypotheses that are either related to timing or to the positions ofevents. A side effect of the DyNAM’s separability in two sub processes is a muchlower computational complexity in the estimation process and thus a much fasterestimation and applicability to larger networks. We conclude that there are no objec-tive criteria that could deem either model superior, but we suggest that researchersshould base their choice of the modeling paradigm (actor-oriented DyNAM ortie-oriented REM) on theoretical assumptions in relation to their hypotheses priorto evaluating fit-based criteria.

The development of the DyNAM framework is tightly connected with thedevelopment of a software package in R (called goldfish), even though in principleresearchers may find use in implementing this statistical approach in their ownsoftware.

Future extensions of the DyNAM framework and the software could, for ex-ample, be concerned with the simultaneous dynamics of one-mode and two-modeinteractions, the development of advanced goodness-of-fit criteria, the introductionof advanced techniques for the analysis of timing that go beyond our time windowapproach, the simultaneous modeling of interpersonal interactions and individualbehavioral actions, and the co-evolution of relational events and network ties.

We believe that the study of social interactions is a timely and exciting topic.Interpersonal interactions are at the core of many sociological theories and they aretightly linked to the dynamics of more stable social network relations. We hope thatDyNAMs will contribute to a deeper understanding of the role of interactions inthe dynamics of complex social networks.

Notes

1 Butts (2008) adds "action type" as a fourth element. This is a natural extension to themost basic case of three elements.

2 Kitts (2014) makes a similar distinction and argues that social network analysis shouldbe concerned with four dimensions of networks: roles, sentiments, interactions, andexchange.

3 Naturally, dependence between events only occurs forward in time.

4 The hazard rate is thus "flat" (Box-Steffensmeier and Jones 2004:22) The process that wedescribe is Markovian. The flatness of the hazard rate can be linked to the "memoryless-ness" of the Markov process.

5 For a detailed introduction of exponential event history models we refer to Box-Steffensmeier and Jones (2004:p.22–25).

6 Note that this is contrary to differences between other actor- and tie-based models, inwhich the model statistics do differ between models. See Block et al. (2016).



7 The DyNAM for undirected events as introduced in Stadtfeld et al. (2017) contains threemodeling steps: (i) who will propose a tie, (ii) to whom the tie is proposed, and (iii)whether this proposal is accepted.

8 The software package is available online at www.social-networks.ethz.ch/research/goldfish.html (retrieved 04/2017). The name refers to the (incorrect) assertion thatgoldfish—like Markov processes—are memoryless. The software has been developed inR (R Core Team 2013). Replication material is available as an online supplement.

9 Stadtfeld and Geyer-Schulz (2011) proposed to combine exponentially weighted effectswith a threshold model that preserves the periodwise stability of the process states.

10 We only use a subsample of the available data between the first and the third surveywave; thus, the friendship network is only updated once.

11 Calculated by the inverse of the "empty" rate 1exp(−13.74) . The result is in seconds.

12 Calculated as 1exp(−13.74+5∗0.59+5∗0.53+5∗0.03+5∗0.26)

13 1exp(−13.74+0.59)/ 1

exp(−13.74) =1

exp(0.59) = 0.55

14 exp(2.07) = 7.9.

15 Calculated as exp(1.52) = 4.6.

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Acknowledgements: Useful feedback and comments that considerably improved thisarticle were graciously provided by James Hollway, by members of the social networkresearch group in Zürich, by participants of the Swiss Networks Workshop in Zürich,and by the 9th International Conference on Social Science Methodology (RC33) inLeicester. Discussions with Alessandro Lomi and Viviana Amati contributed to theformulation of the idea of network mechanisms that operate on different time scales—they refer to this idea as "process time."

Christoph Stadtfeld: Department of Humanities, Social and Political Sciences, ETH Zürich.E-mail: [email protected].

Per Block: Department of Humanities, Social and Political Sciences, ETH Zürich.E-mail: [email protected].


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